Revise Greenhouse effect for IB Physics (B.2)

A black body is an idealised object that absorbs all radiation falling on it and, being the perfect absorber, is also the best possible emitter at any given temperature. Its emission spectrum — a graph of radiated intensity against wavelength — has a characteristic humped shape that depends only on temperature, not on what the body is made of. The Sun, stars and planets are all modelled as black bodies to good approximation. As a body gets hotter two things happen at once: the peak of the spectrum shifts to shorter wavelengths, and the area under the whole curve (the total power radiated) grows rapidly. Those two behaviours are captured by Wien's displacement law and the Stefan–Boltzmann law respectively.

Wien's displacement law states that the wavelength at which the emission spectrum peaks is inversely proportional to the absolute temperature of the body.

$\lambda_{max} = \frac{2.90 \times 10^{-3}}{T}$

where $\lambda_{max}$ is the peak wavelength in metres, $T$ is the absolute temperature in kelvin, and $2.90 \times 10^{-3}$ m K is Wien's constant (data booklet). Because the relationship is inverse, doubling the temperature halves the peak wavelength. This law is what lets astronomers read a star's surface temperature straight off the colour of its light, and it is why a warming filament glows first dull red, then orange, then white as its peak marches from long to short wavelengths.

The Stefan–Boltzmann law gives the total power radiated by a black body across all wavelengths. For a body of surface area $A$ at absolute temperature $T$, the radiated power (often called luminosity $L$ for a star) is $L = \sigma A T^4$, with $\sigma = 5.67 \times 10^{-8}$ W m⁻² K⁻⁴. The fourth-power dependence is dramatic: doubling the temperature multiplies the power by $2^4 = 16$. Real surfaces radiate a little less than a perfect black body, and this is accounted for by the emissivity $e$ (between 0 and 1), giving $L = e\sigma A T^4$; a perfect black body has $e = 1$. On the incoming side, not all sunlight reaching a planet is absorbed — the albedo $\alpha$ is the fraction reflected straight back to space, so only a fraction $(1-\alpha)$ is available to warm the planet. Keep the two ideas distinct: albedo governs reflection of incoming radiation, emissivity governs emission of outgoing radiation.

The Sun's power spreads out over an ever-larger sphere as it travels, so the intensity (power per unit area) falls off as the inverse square of distance: $I = \dfrac{P}{4\pi d^2}$. Evaluated at Earth's orbital distance, this intensity is called the solar constant, $S \approx 1.36 \times 10^{3}$ W m⁻² — the power per square metre arriving at the top of the atmosphere on a surface facing the Sun. But a planet does not present its whole surface to this beam. It intercepts sunlight only over its cross-sectional disc of area $\pi R^2$ (the shadow it casts), while it radiates its own infrared from its entire spherical surface $4\pi R^2$. That mismatch — catch over a disc, emit over a sphere — introduces the crucial factor of 4 into the energy balance.

$S = \frac{L_{Sun}}{4\pi d^2}$

If a planet's average temperature is steady, it must be in radiative equilibrium: the power it absorbs from the Sun equals the power it radiates back to space. The power absorbed is the solar constant times the intercepting disc, reduced by the reflected fraction: $P_{in} = S(1-\alpha)\pi R^2$. The power radiated, from the whole sphere, is $P_{out} = e\sigma(4\pi R^2)T^4$. Setting $P_{in} = P_{out}$, the $\pi R^2$ cancels — so the equilibrium temperature does not depend on the planet's size, only on the incoming intensity, its albedo and its emissivity.

$S(1-\alpha)\pi R^2 = e\sigma (4\pi R^2) T^4 \quad\Longrightarrow\quad T = \sqrt[4]{\frac{S(1-\alpha)}{4e\sigma}}$

For a black-body Earth ($e = 1$) this predicts about 255 K, some 33 K below the observed 288 K. The atmosphere makes up the difference, and that is the greenhouse effect.

The atmosphere is largely transparent to incoming short-wave sunlight, which passes through and is absorbed at the surface. The warmed surface then radiates long-wave infrared (its Wien peak lies in the IR because it is cool). Greenhouse gases — chiefly water vapour (H₂O), carbon dioxide (CO₂), methane (CH₄) and nitrous oxide (N₂O) — absorb this outgoing infrared. An absorbed IR photon raises a molecule to a higher vibrational energy state; the excited molecule then re-emits infrared in a random direction, so a significant fraction is radiated back down towards the surface. This returned radiation reduces the net rate of energy loss to space, and the surface must settle at a higher temperature before incoming and outgoing power balance once more. The molecular requirement is that the vibration changes the molecule's electric dipole moment; symmetric diatomic molecules like N₂ and O₂ cannot do this and so are not greenhouse gases, which is why the bulk of the atmosphere is IR-transparent.

The natural greenhouse effect described above is essential: without it Earth would sit near −18°C and be largely frozen. The enhanced greenhouse effect is the additional warming caused by human activities that raise greenhouse-gas concentrations — principally CO₂ from burning fossil fuels and deforestation, along with extra methane from agriculture and fossil-fuel extraction. Adding greenhouse gases increases the absorption of outgoing infrared, shifting the energy balance so that the surface warms until balance is restored at a higher temperature. Because CO₂ persists in the atmosphere for centuries and, unlike water vapour, is not controlled by temperature, it is treated as the primary driver of recent climate change even though water vapour is more abundant. Be precise with the terminology: the greenhouse effect is natural and beneficial, while the enhanced greenhouse effect is the human-driven intensification behind global warming.

B.2 mixes calculation with explanation, and the explanation marks are awarded analytically — each distinct valid physics point is worth one mark. Method marks (M) credit correct reasoning, answer marks (A) credit a correct conclusion, and error-carried-forward (ECF) means a wrong value early on does not cost you the marks that follow, provided your method is written down. Study how each mark below is tied to a specific, named idea rather than to loose phrasing.

The radiation laws in this topic reach well beyond climate. Wien's law and the Stefan–Boltzmann law are the tools for reading a star's temperature and luminosity in astrophysics; the inverse-square intensity relation reappears wherever power spreads from a source, from sound to gravitation. Master the energy-balance habit — absorbed power in over the disc, radiated power out over the sphere, set them equal — and you have a template that solves planetary-temperature problems for any world, atmosphere or none.